84 G. FLEURI. 



sufficient, to introduce a certain kind of magnitude into calcula- 

 tion is to be able to define in a precise manner the equality and 

 the um of two magnitudes of that kind. For every kind of 

 magnitude appropriate definitions are necessary which depend on 

 the nature of the kind of magnitude considered. 



With regard to absolute magnitudes first, the definitions are 

 always given in such a manner, that, if two magnitudes are equal 

 their tensors are equal and that the tensor of the sum equal the 

 sum of the tensors, so that in that case the study of operations on 

 absolute magnitudes come to the study of operations on numbers. 

 This study is the aim of arithmetic. 



"With regard to directive plane magnitudes, the definitions are 

 always given in such a manner that two equal magnitudes have 

 equal tensors and arguments, and that the sum be obtained by 

 the rule of the parallelogram. The study of operations on those 

 magnitudes is therefore the aim of algebra. 



We come now to a third series of magnitudes the one which 

 can have any direction in space. As long as we consider their 

 tensors only, we can operate on them with algebra, but when we 

 come to consider those magnitudes in direction as well, other 

 processes of calculation are necessary and we are then naturally 

 introduced to the calculus of quaternions. 



But before coming to the consideration of quaternions, an idea 

 at first strikes one forcibly. Why could not algebra deal with 

 space magnitudes 1 The reason of it is the impossibility of keep- 

 ing in the multiplication of space quantities the property of com- 

 mutativity which is, as we have explained above, fundamental in 

 algebra. 



However it may be a question whether it would not be possible 

 to find another system of calculation of space quantities leaving 

 untouched the commutativity of multiplication. We will try to 

 demonstrate that " a priori " the thing is not possible, that is to 

 say that algebraical calculations with its laws does not allow us 

 to deal with any quantity but complex quantities. 



